Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T21:04:13.082Z Has data issue: false hasContentIssue false
  • English
  • Français

Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Parabolic equations and systems

Published online by Cambridge University Press:  07 February 2017

Necibe Tuncer  and
Anotida Madzvamuse
Necibe Tuncer*
Affiliation:
Department of Mathematics, Florida Atlantic University, 777 Glades Road, Boca Raton, Fl 33431, USA
Anotida Madzvamuse*
Affiliation:
University of Sussex, School of MPS, Department of Mathematics, BN1 9QH, Brighton, UK
*
*Corresponding author. Email addresses:ntuncer@fau.edu (N. Tuncer), a.madzvamuse@sussex.ac.uk (A. Madzvamuse)
*Corresponding author. Email addresses:ntuncer@fau.edu (N. Tuncer), a.madzvamuse@sussex.ac.uk (A. Madzvamuse)
Get access

Abstract

The focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as the activator-activator and short-range inhibition; long-range activation.

Keywords

Time-dependent projection operatorsprojected finite elementsnon-autonomous partial differential equationsreaction-diffusion systemsevolving surfacesTuring diffusively-driven instabilitypattern formation

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35K57: Reaction-diffusion equations 35K58: Semilinear parabolic equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 718 - 747
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Barreira, R., Elliott, C.M., and Madzvamuse, A.. The surface finite element method for pattern formation on evolving biological surfaces. J. Math. Biol., 63(6):1095–1119, 2011.CrossRefGoogle ScholarPubMed
[2] Blazakis, K., Aldasoro, C. R., Venkataraman, C., Styles, V., and Madzvamuse, A.. An optimal control approach for neutrophil cell motility. In preparation.Google Scholar
[3] Caginalp, G.. Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. Phys. Rev. A (3), 39(11):5887–5896, 1989.CrossRefGoogle ScholarPubMed
[4] Chaplain, M. A. J., Ganesh, M., and Graham, I. G.. Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J. Math. Biol., 42(5):387–423, 2001.CrossRefGoogle ScholarPubMed
[5] Dagdas, Y. F., Yoshino, K., Dagdas, G., Ryder, L. S., Bielska, E., Steinberg, G., and Talbot, N. J.. Septin-mediated plant cell invasion by the rice blast fungus, magnaporthe oryzae. Science, 336(6088):1590–1595, 2012.CrossRefGoogle ScholarPubMed
[6] Deckelnick, K., Dziuk, G., and Elliott, C. M.. Computation of geometric partial differential equations and mean curvature flow. Acta Numer., 14:139–232, 2005.CrossRefGoogle Scholar
[7] Donna, A. and Helzel, C.. A finite volume method for solving parabolic equations on logically Cartesian curved surface meshes. SIAM J. Sci. Comput., 31(6):4066–4099, 2009.Google Scholar
[8] Dziuk, G. and Elliott, C. M.. Finite elements on evolving surfaces. IMA J. Numer. Anal., 27(2):262–292, 2007.CrossRefGoogle Scholar
[9] Dziuk, G. and Elliott, C. M.. Surface finite elements for parabolic equations. J. Comput. Math., 25(4):385–407, 2007.Google Scholar
[10] Dziuk, G. and Elliott, C. M.. Surface finite elements for parabolic equations. J. Comput. Math., 25(4):385–407, 2007.Google Scholar
[11] Dziuk, G. and Elliott, C. M.. Eulerian finite element method for parabolic PDEs on implicit surfaces. Interfaces Free Bound., 10(1):119–138, 2008.CrossRefGoogle Scholar
[12] Dziuk, G. and Elliott, C. M.. An Eulerian approach to transport and diffusion on evolving implicit surfaces. Comput. Vis. Sci., 13(1):17–28, 2010.CrossRefGoogle Scholar
[13] Dziuk, G. and Elliott, C. M.. Finite element methods for surface PDEs. Acta Numer., 22:289–396, 2013.CrossRefGoogle Scholar
[14] Elliott, C. M., Stinner, B., Styles, V., and Welford, R.. Numerical computation of advection and diffusion on evolving diffuse interfaces. IMA J. Numer. Anal., 31(3):786–812, 2011.CrossRefGoogle Scholar
[15] Gierer, A. and Meinhardt, H.. Theory of biological pattern formation. Kybernetik, 12(1):30–39, 1972.CrossRefGoogle ScholarPubMed
[16] Greer, J. B., Bertozzi, A. L., and Sapiro, G.. Fourth order partial differential equations on general geometries. J. Comput. Phys., 216(1):216–246, 2006.CrossRefGoogle Scholar
[17] Hetzer, G., Madzvamuse, A., and Shen, W.. Characterization of Turing diffusion-driven instability on evolving domains. Discrete Contin. Dyn. Syst., 32(11):3975–4000, 2012.CrossRefGoogle Scholar
[18] Hieber, S. E. and Koumoutsakos, P.. A Lagrangian particle level set method. J. Comput. Phys., 210(1):342–367, 2005.CrossRefGoogle Scholar
[19] Kadirkamanathan, V., Anderson, S., Billings, S., Zhang, X., and Holmes, G.. The neutrophil's eye-view: Inference and visualisation of the chemoattractant field driving cell chemotaxis in vivo. PLoS ONE, 7(4):e35182, 2012.CrossRefGoogle ScholarPubMed
[20] Kondo, S. and Asai, R.. A reaction-diffusion wave on the skin of the marine angelfish pomacanthus. Nature, 376(6543):765–768, 1995.CrossRefGoogle ScholarPubMed
[21] Lakkis, O., Madzvamuse, A., and Venkataraman, C.. Implicit–explicit timestepping with finite element approximation of reaction–diffusion systems on evolving domains. SIAM Journal on Numerical Analysis, 51(4):2309–2330, 2013.CrossRefGoogle Scholar
[22] Lefèvre, J. and Mangin, J.-F.. A reaction-diffusion model of human brain development. PLoS Comput. Biol., 6(4):e1000749, 2010.CrossRefGoogle ScholarPubMed
[23] Liaw, S. S., Yang, C. C., Liu, R. T., and Hong, J. T.. Turing model for the patterns of lady beetles. Phys. Rev. E., 64:041909, 2001.CrossRefGoogle ScholarPubMed
[24] Macdonald, C. B., Merriman, B., and Ruuth, S. J.. Simple computation of reaction-diffusion processes on point clouds. Proc. Natl. Acad. Sci. USA, 110(23):9209–9214, 2013.CrossRefGoogle ScholarPubMed
[25] Macdonald, C. B. and Ruuth, S. J.. The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput., 31(6):4330–4350, 2009/10.CrossRefGoogle Scholar
[26] Madzvamuse, A.. Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. J. Comput. Phys., 214(1):239–263, 2006.CrossRefGoogle Scholar
[27] Madzvamuse, A., Gaffney, E. A., and Maini, P. K.. Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains. J. Math. Biol., 61(1):133–164, 2010.CrossRefGoogle ScholarPubMed
[28] Madzvamuse, A., Ndakwo, H. S., and Barreira, R.. Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion. Discrete and Continuous Dynamical Systems - Series A, 34(4):2133–2170, 2016.Google Scholar
[29] Madzvamuse, A., Wathen, A. J., and Maini, P. K.. A moving grid finite element method applied to a model biological pattern generator. J. Comput. Phys., 190(2):478–500, 2003.CrossRefGoogle Scholar
[30] Meinhardt, H.. The Algorithmic Beauty of Sea Shells. The Virtual Laboratory. Springer-Verlag, Berlin, 1995.CrossRefGoogle Scholar
[31] Meir, A. J. and Tuncer, N.. Radially projected finite elements. SIAM J. Sci. Comput., 31(3):2368–2385, 2009.CrossRefGoogle Scholar
[32] Murray, J. D.. Mathematical biology. II, volume 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York, third edition, 2003. Spatial models and biomedical applications.CrossRefGoogle Scholar
[33] Osher, S. and Fedkiw, R.. Level set methods and dynamic implicit surfaces, volume 153 of Applied Mathematical Sciences. Springer-Verlag, New York, 2003.CrossRefGoogle Scholar
[34] Prigogine, I. and Lefever, R.. Symmetry breaking instabilities in dissipative systems II. J. Chem. Phys., 48:1695–1700, 1968.CrossRefGoogle Scholar
[35] Ramms, L., Fabris, G., Windoffer, R., Schwarz, N., Springer, R., Zhou, C., Lazar, J., Stiefel, S., Hersch, N., Schnakenberg, U., Magin, T. M., Leube, R. E., Merkel, R., and Hoffmann, B.. Keratins as the main component for the mechanical integrity of keratinocytes. Proceedings of the National Academy of Sciences, 110(46):18513–18518, 2013.CrossRefGoogle ScholarPubMed
[36] Schnakenberg, J.. Simple chemical reaction systems with limit cycle behavior. J. Theoret. Biol., 81(3):389–400, 1979.CrossRefGoogle Scholar
[37] Sethian, J. A.. Level set methods and fast marching methods, volume 3 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, second edition, 1999. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science.Google Scholar
[38] Tuncer, N.. A novel finite element discretization of domains with spheroidal geometry. Ph.D. Dissertation, Auburn University Libraries, 2007.Google Scholar
[39] Tuncer, N., Madzvamuse, A., and Meir, A. J.. Projected finite elements for reaction-diffusion systems on stationary closed surfaces. Applied and Numerical Mathematics, 96:45–71, 2015.CrossRefGoogle Scholar